which is nothing else but the Green function \(G({\bf x}_1,{\bf x}_2)\)  of the Laplace operator on the domain \({\bf D}\) . Note however that mathematically \(V({\bf x})\) 
is rather subtle (e.g. the sum in (REF ) does not converge pointwise and fails in general
to be an element of the Hilbert space \(\texttt {H}\)  ).
Because of this and other subtleties an extra mathematical care is needed to define the object fully rigorously,
see references in [1]}. The physicists however work with such an object without further ado, and we finish this section
by two simple but important examples. In the first example we deal with the GFF on a one-dimensional domain, the interval \({\bf D}=[0,1]\) .
The Laplacian in one dimension is simply \(\Delta =-\frac{d^2}{dx^2}\)  and the eigenfunctions/eigenvalues of the Dirichlet problem are given by \(e_n(x)=\sqrt{2}\sin {n\pi x}, \lambda _n=\pi ^2n^2\)  so that the GFF in this particular case is given by a random Fourier series
\(V(x)=\sum _{n=1}^{\infty } \zeta _n \frac{\sqrt{2}}{\pi n} \sin {n\pi x}\)  (compare with the periodic \(1/f\)  noise in the end of this Appendix). The corresponding
Green function can be easily found to be given by \(G(x_1,x_2)=x_1(1-x_2)\)  for \( x_2>x_1\)  and \(G(x_1,x_2)=x_2(1-x_1)\)  for \( x_2<x_1\) .
One immediately recognizes that the one-dimensional version of the GFF for such a domain coincides with the version of the
Brownian motion called Brownian bridge, which is conditioned to return to the origin after a given time.
